2 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
variety
V
defined over
K
has a K-rational point, provided that
V
has a non-
singular point over each real closure of
K.
The latter condition on
Vis
equivalent
to the following one: the function field K(V) of V is a totally real extension of
K, that is, every ordering on K extends to K(V).
The field Qltr of all totally real algebraic numbers is the fixed field of all
involutions in the absolute Galois group G(Ql) of Ql. From Pop
[P],
Qltr is PRC.
By Weissauer's theorem
[FJ,
Proposition 12.4] every proper finite extension of
Qltr is Hilbertian. Also, by Prestel's extension theorem
[P2,
Theorem 3.1] every
algebraic extension of Qltr is PRC. Hence, the absolute Galois group of a proper
finite extension of Qltr is known
[FV3].
The field Qltr, however, is not Hilbertian.
For example, Z
2
- (
a2
+
1) is reducible over Qltr for every a
E
Qltr.
In
[FHV]
we cover also the case of Qltr. The key observation is that Qltr
satisfies a certain weakening of the Hilbertian property. This allows specializing
Galois extensions of
K(x)
whose Galois group is generated by real involutions
to obtain Galois extensions of K with the same Galois group. As a result we
determine the absolute Galois group of Qltr.
In the present paper we extend the methods and results of
[FV3].
This
solves regular embedding problems over a PRC field
K
so that the orderings
of K extend to prescribed subfields (Theorem 5.2). Thus, Theorem 5.3 gives
new information about the absolute Galois group of the field
K(x)
of rational
functions over K .
The first 5 sections setup the proof of Theorem 5.2. We need an approximation
theorem for varieties over PRC fields (section 1), supplements about the moduli
spaces (section 2), group-theoretic lemmas (section 3), and the determination of
the real involutions in Galois groups over JR(x) (section 4).
In section 6 we define the concept of totally real Hilbertian, and show that Qltr
has this property. Section 7 combines these results to determine the absolute
Galois group of a countable totally real Hilbertian PRC field satisfying these
properties: it has no proper totally real algebraic extensions; and its space of
orderings has no isolated points. This group is the free product of groups of
order 2, indexed by the Cantor set
Xw
(Theorem 7.6). In particular, G(Qltr) =
Aut(Q/Qltr) is isomorphic to this group.
As a corollary we introduce the notion of real Frobenius fields: Qltr is an
example (Corollary 8.3). Following the Galois stratification procedure of
[FJ,
Chap. 25] and
[HL]
we show that the elementary theory of real Frobenius fields
allows elimination of quantifiers in the appropriate language. In particular, Qltr
is primitive recursively decidable (Theorem 10.1). On the other hand the ring
of integers of Qltr is undecidable (A. Prestel pointed us to Julia Robinson's proof
of this
[R2].)
Thus we obtain a natural example of an undecidable ring with a
decidable quotient field. Compare this with the possibility that Ql is decidable
(cf.
Robinson
[Rl,
p. 951]). Furthermore, we give (Corollary 10.5) a system of
axioms for the theory of Qltr.
Affirmations. We are grateful to Moshe Jarden for numerous suggestions
Previous Page Next Page